The stuttering manner in which Manchester United took the three points provided further confirmation of the surgery required from David Moyes. This season's vintage can be shaky at the back, sluggish once the centre circle is passed and seriously lacking in ideas when the opposition crowd them, as Cardiff City did at times on Tuesday night.

That was despite the debut of Juan Mata, following his transfer from Chelsea, though when the Spaniard finds the smooth rhythm with which he can run matches and win them United may become a more potent force than the team that still trail a Champions League position by six points, thanks to Liverpool's win in the Merseyside derby.

Moyes welcomed victory but concurred that his side must get better. "Has my mood improved? Yeah, because I know I have got some really good players coming back but of course I want to be more convincing, I want to win better and play much better," he said. "Don't think because we have won tonight I have got a smile and am happy, I'm happy because we have got the three points but for where I want to go and the vision I have got for how I want my team to go, I have got a long way to go."

Alongside Mata, the club record £37.1m buy, the players of which the manager spoke were Robin van Persie, who last featured on 10 December because of a thigh problem, and Wayne Rooney, named as a substitute after the abductor injury that had caused the England international to miss five games.

For the returning old boy Ole Gunnar Solskjaer, who retired seven years ago, there was a tumultuous reception when the Cardiff manager strode out to take up his seat on the bench.

United started brightly enough and were ahead after six minutes after a move begun by Mata. The Spaniard swept a pass to the left which Ashley Young left for Patrice Evra. From him it went to Van Persie and then to Young, who curled in a cross from which Antonio Valencia headed against the bar. Van Persie reacted to the rebound and, although David Marshall saved his initial header, the Dutchman made no mistake with a second for his first goal since the winner against Arsenal at Old Trafford on 10 November.

Moyes said: "That's what Robin has done and that's what we have missed – having goalscorers who can change games. Goals change games and some of our play has deserved goals and we have not finished it off."

The champions also need to replenish central midfield and the way the 22-year-old Jordan Mutch ghosted past Ryan Giggs, now a greying 40, rammed home the point.

Moyes also has doubts about his defence. On more than one occasion, the centre-back pairing of Chris Smalling and Jonny Evans have struggled in the air this season, and when Craig Noone nearly profited from a Cardiff corner after the Northern Ireland defender missed a header it brought the manager out of his seat to make clear his displeasure.

By the break another of the side's problems had surfaced: the sideways, half-paced possession football that makes United too static. Moyes hopes that Mata, Rooney, Van Persie and Adnan Januzaj, who came on as a second half replacement, will make the team more fluid.

"Now we have got competition and I can use them at different times, I can change them within games if I need to do that," Moyes said. "Wayne has dropped back into midfield in some games at different times. We have got options and I want to be able to do that, change within the game and Juan Mata more important creates, scores and you can't get enough of them, especially here at Old Trafford."

The sense that Cardiff might swarm forward to equalise at any moment was heightened when Kim Bo-kyung, on for Mutch in the 52nd minute, danced down the right of United's area and fired in a cross that Fraizer Campbell was allowed to meet first. This followed Declan John's dangerous run down the left as the visitors enjoyed time a sustained spell in and around the opposing area.

This all meant that Young's strike brought extra relief when it arrived soon afterwards. Mata was the provider, setting up the winger, who cut inside from the left and beat Marshall with ease. To add to the United fans' satisfaction Rooney came on for Van Persie in the 63rd minute, slotting into the lone central berth to leave Mata in is his favoured No10 position.

The defeat left Cardiff rooted to the bottom. "It's hard when you concede a goal within six minutes," Solskjaer said. "We contained them and played better but didn't have the urgency or threat to create big chances. We did cause them some problems but didn't have enough."